Good point about the PMI. There are ways to avoid it, if you qualify for certain government programs. There are also ways to borrow the down payment (separate from the mortgage), which avoids the PMI, but may cost you more in the long run with the interest on the other loan.

Points are basically pre-paying some interest. You can pay more points up front (each point is 1% of the loan value) and your interest rate goes down. Some lenders also offer "negative points" where you get cash back at closing (likely will just offset your closing costs) and your interest rate is higher. Whether you should pay points or not depends on many factors, like how long you plan to stay in the home and how much cash you have on hand.

Something I forgot to mention before is closing costs. The buyer is responsible for a ton of costs upfront. Your mortgage company will want an appraisal. Unless you're very knowledgeable and handy, you probably want a home inspection (which may also be required by your mortgage company). There are a ton of required filing fees to various government entities. Your mortgage company will charge various fees (that you can maybe negotiate, probably better to just shop around). Buying and selling real estate is expensive, which is why you shouldn't buy a house if you don't plan to be there for a while. I've always seen 5 years as a general rule-of-thumb - if you'll be somewhere less than 5 years, you're better off renting.